Question 1204179
**1. Find the Standard Deviation of Y**

* **Y = a + bX** 
* **σY = |b| * σX** 
    * The absolute value of 'b' is used because standard deviation must be non-negative.

**2. Find the Covariance of X and Y**

* **Cov(X, Y) = Cov(X, a + bX)** 
* **Cov(X, Y) = Cov(X, a) + Cov(X, bX)** 
* **Cov(X, Y) = 0 + b * Var(X)** 
    * Cov(X, a) = 0 because the covariance between a random variable and a constant is zero.
    * Cov(X, bX) = b * Var(X) 

* **Cov(X, Y) = b * σX²**

**3. Calculate the Correlation Coefficient (ρXY)**

* **ρXY = Cov(X, Y) / (σX * σY)** 
* **ρXY = (b * σX²) / (σX * |b| * σX)** 
* **ρXY = b / |b|**

**4. Determine the Sign of ρXY**

* **If b > 0:** 
    * ρXY = b / b = 1 
    * This indicates a perfect positive linear relationship between X and Y.

* **If b < 0:**
    * ρXY = b / (-b) = -1 
    * This indicates a perfect negative linear relationship between X and Y.

**Therefore:**

* If b < 0, ρXY = -1
* If b > 0, ρXY = 1

This demonstrates that the correlation coefficient between X and Y is perfectly positively or negatively correlated depending on the sign of the slope (b) in the linear relationship Y = a + bX.